Theorem atnaiana 43166

Description: Given a, it is not the case a implies a self contradiction. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypothesis

Ref	Expression
atnaiana.1	⊢ 𝜑

Assertion

Ref	Expression
atnaiana	⊢ ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))

Proof of Theorem atnaiana

Step	Hyp	Ref	Expression
1		atnaiana.1	. . . 4 ⊢ 𝜑
2	1	bitru 1546	. . 3 ⊢ (𝜑 ↔ ⊤)
3		pm3.24 405	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ 𝜑)
4	3	bifal 1553	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
5	2, 4	aifftbifffaibif 43164	. 2 ⊢ ((𝜑 → (𝜑 ∧ ¬ 𝜑)) ↔ ⊥)
6	5	aisfina 43141	1 ⊢ ¬ (𝜑 → (𝜑 ∧ ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-fal 1550

This theorem is referenced by:  ainaiaandna  43167  confun5  43186